DER 5 Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities
Price and value are two separate ideas for a forward commitment, and keeping them apart is the whole foundation of this reading. The forward price is the rate at which the two counterparties agree to exchange the underlying (or cash) on a future date. It is locked in when the contract is struck and never moves afterward. The value is what the contract is worth to a counterparty at a given moment, which is the gain or loss that would be realized if the position were closed out right then.
A forward, future, or swap starts life with a value of zero to both sides, once trading costs and counterparty credit are set aside. No cash changes hands at inception. Even so, the agreed forward price is not the current spot price: it already carries the opportunity cost of holding a long cash position, measured by the risk-free rate, plus any cost or benefit of owning the underlying. As time passes and the spot price and other inputs move, the contract acquires a mark-to-market (MTM) value. That value is symmetric, so whatever the seller gains, the buyer loses by the same amount, and the reverse. One structural difference is worth flagging: an exchange clearinghouse settles futures MTM changes in cash every day, whereas a forward is normally settled once, at maturity.
Zero value at initiation
Consider an equity forward in which AMY Investments agrees to buy 1,000 Airbus shares one year out at a forward price of EUR30 per share. With a spot price of EUR29.70 and a one-year risk-free rate of 1.00 percent, that EUR30 quote meets the no-arbitrage test at the outset, leaving it as neither asset nor liability for either party.
Settlement at maturity
A forward has a symmetric payoff, so at expiry the long position collects the difference between the terminal spot price and the fixed forward price, while the short position collects the mirror image. Because settlement occurs only at expiry, the worth at that date is exactly the settlement amount viewed from each side.
| Relationship at T | Long position | Short position |
|---|---|---|
| Spot above forward | Positive gain | Negative, a loss |
| Spot below forward | Negative, a loss | Positive gain |
| Spot equals forward | Zero | Zero |
Long value equals terminal spot minus forward price; short value is its negative.
The Viswan Family Office (VFO) holds 10,000 non-dividend-paying shares of Biomian Limited, a Mumbai biotech firm, at a spot price of INR295 per share. VFO agrees to sell 1,000 of those shares forward to a financial intermediary at INR300.84 each, settling in six months. At inception the contract value to both parties is zero.
(1) Spot INR287. The buyer (long) collects 287 minus 300.84, a value of minus INR13.84. VFO (short) collects 300.84 minus 287, a value of plus INR13.84.
(2) Spot INR312. The buyer collects 312 minus 300.84, a value of plus INR11.16. VFO collects 300.84 minus 312, a value of minus INR11.16. The seller does well when the spot ends below the locked forward price, and poorly when it ends above.
Once the contract is running, the passage of time and shifts in the spot price keep changing its worth. For an underlying with no cost or benefit of ownership, no-arbitrage forces the forward price to equal the future value of the spot price grown at the risk-free rate. To find the MTM value part way through, we compare the current spot price against the present value of that fixed forward price, discounted back over the time still left.
With the current spot price written as the value at time t, the mark-to-market value to the long position over the life of the contract is the spot price minus the discounted forward price.
Picture the present value of the forward price as a line that starts at the current spot price when the contract begins and climbs to the full forward price at maturity, rising at the risk-free rate. At any moment the vertical distance between the actual spot price and that line is the long position’s gain or loss.
The sign of the value depends on where the current spot sits relative to that discounted forward price. When spot is above it, the long gains and the short loses; when spot is below it, the positions reverse; when the two are equal, neither side has a gain or loss.
As in Example 1, VFO enters a six-month forward to sell Biomian shares at a forward price of INR300.84, with the current spot price at INR295.
2. At the instant of inception t equals zero, so discounting the forward price back over the full term returns the original spot price. The seller’s value therefore collapses to the initial spot minus the new spot, 295 minus 325, which is minus INR30, a loss for the short position as the price runs against it.
Again VFO is short the six-month Biomian forward at INR300.84. The starting spot is INR295 and the risk-free rate is 4 percent.
1. With the forward price INR300.84 discounted at 4 percent over the remaining quarter, 300.84 times (1.04) to the power minus 0.25, then minus 285, the result is an INR12.90 gain.
2. Repeating at 8 percent, 300.84 times (1.08) to the power minus 0.25, minus 285, gives an INR10.11 gain. The seller’s gain has shrunk by INR2.79. A higher risk-free rate raises the opportunity cost of holding cash and lowers the present value of the forward price, which trims the short position’s gain.
The examples so far assumed the underlying threw off no cash. Most real assets carry a mix of benefits and costs of ownership, known together as the cost of carry: income (I) such as dividends or coupons is a benefit, while storage, insurance, and similar outlays (C) are costs. Netting these present values into the spot price before compounding at the risk-free rate gives the forward price that satisfies no-arbitrage.
Income lowers the forward price because the holder of the asset receives it and the forward buyer does not, while carrying costs raise it. During the life of the contract the long position’s MTM value is the current spot adjusted for whatever costs and benefits remain from now through maturity, minus the discounted forward price.
Hightest Capital agreed to deliver 1,000 Unilever shares in six months at a forward price of EUR50,631.10 in total, or EUR50.6311 per share. Unilever pays a quarterly dividend of EUR0.30 three months after inception and again at maturity, and the risk-free rate is 5 percent, flat.
Foreign exchange forwards
A currency forward is the same idea with two interest rates in play. Both the spot and forward quotes are stated as units of a price currency (the foreign side, f) per one unit of a base currency (the domestic side, d). The relationship between spot and forward reflects the gap between the foreign and domestic risk-free rates, shown here in continuous-compounding form, and the MTM value at any time discounts the forward price at the current rate differential.
At inception the currency with the lower risk-free rate trades at a forward premium, meaning fewer units of the other currency buy one unit of it forward, while the higher-rate currency trades at a forward discount. A wider differential, that is a larger foreign-minus-domestic rate gap, pushes the price (foreign) currency to depreciate on a forward basis and the base (domestic) currency to appreciate.
Rook Point Investors LLC is long a one-year USD/EUR forward, agreeing to buy EUR1,000,000 for USD1,201,000 in one year. At inception the USD/EUR spot is 1.192 (USD1.192 per EUR1), the one-year USD rate is 0.50 percent, and the one-year EUR rate is minus 0.25 percent. Here USD is the price (foreign) currency and EUR is the base (domestic) currency.
Rook Point is instead short a six-month ZAR/EUR forward, agreeing to sell ZAR and buy EUR at a forward price of 17.2506. The spot is 16.909, the South African rate is 3.5 percent, and the EUR rate is minus 0.5 percent. As the seller, Rook Point values the position as the present value of the forward price minus the spot: 17.2506 times e to the power minus (0.035 minus minus 0.005) times 0.5, which comes back to 16.909. If the ZAR/EUR spot appreciates to 16.5, the value becomes 16.909 minus 16.5, a loss of 0.4090, because the seller has locked in a sale at the original rate while the market has moved against that position.
So far a single constant risk-free rate stood in for the opportunity cost of holding the underlying. Interest rates are different: they have a term structure, so a distinct rate applies to each time-to-maturity. The same principle carries over to other variables that carry a term structure, including credit spreads, implied volatility, and foreign exchange, where two rate structures interact. Interest rate derivatives are priced off the value and yield of single future cash flows, so the first job is to strip coupon-bond yields into pure single-cash-flow rates.
A zero rate (or spot rate) is the yield on a zero-coupon bond for a given maturity. Starting from the shortest bond and working outward, each new zero rate is solved using the ones already found, a process known as bootstrapping or forward substitution.
Three recently issued annual-coupon government bonds trade at a discount to face value. Their yields-to-maturity come from a standard bond-pricing solve (for instance, the three-year bond gives 3.9703 percent).
| Years to maturity | Annual coupon | Price | YTM | Zero rate |
|---|---|---|---|---|
| 1 | 1.50% | 99.125 | 2.3960% | 2.3960% |
| 2 | 2.50% | 98.275 | 3.4068% | 3.4197% |
| 3 | 3.25% | 98.000 | 3.9703% | 4.0005% |
The price counterpart of a zero rate is the discount factor: the present value today of one currency unit received on a future date. It can also be read as the price of a zero-coupon cash flow.
Applying it to the three zero rates gives one-year, two-year, and three-year discount factors of 0.976601, 0.934961, and 0.888982. Because a zero-coupon bond is just a known future amount, no-arbitrage says it must trade at that amount times the right discount factor; if it does not, a riskless profit appears.
Take a two-year risk-free zero rate of 3.42 percent and a two-year zero-coupon bond of GBP100 face quoted at GBP92.45. The no-arbitrage price is GBP100 divided by (1.0342) squared, which is GBP93.4955, so the bond is cheap. A trader borrows GBP92.45 at 3.42 percent and buys the bond, an opening cash flow of zero. Two years on, the bond pays GBP100 and the loan repayment is GBP92.45 times (1.0342) squared, which is GBP98.88, leaving a riskless profit of GBP1.12.
A forward interest rate contract fixes a rate for a future window, so its name has to state both when the window opens and how long it runs. A two-year forward that references a three-year rate, starting at the end of year two and maturing at the end of year five, is a “2y3y” forward rate. The first figure is the start point, the second is the tenor of the underlying rate. For short-term market reference rates (MRRs) the same tags are quoted in months, so a 3m6m rate is a six-month MRR beginning three months out and maturing nine months from today.
An implied forward rate is the breakeven reinvestment rate connecting a near-term zero-coupon bond with a longer one. It is the future-period rate at which an investor ends up equally well off whether they invest to the near date and roll over at that rate, or invest straight through to the far date. Because the two strategies must return the same amount, the forward rate is pinned down by no-arbitrage.
Using the zero rates from the earlier bond set, z₁ equals 2.396 percent and z₂ equals 3.4197 percent, an investor weighs two ways to cover a two-year horizon: put USD100 in for one year and roll the proceeds at the one-year rate one year forward (the 1y1y rate, IFR₁,₁), or lock in the two-year zero rate for the full period.
2. Using the two- and three-year rates, (1.034197) squared times (1 plus IFR₂,₁) equals (1.040005) cubed, which gives IFR₂,₁ equal to 5.1719 percent.
The general link between two spot rates and the forward rate between them uses a shorter bond maturing in A periods and a longer bond maturing in B periods; the implied forward rate runs from period A to period B, with a tenor of B minus A.
Stringing these forward rates together builds a forward curve over the same horizon. When the spot (zero) curve slopes upward, each forward rate sits above the spot rate for the corresponding period, as the derived rates 4.4536 percent and 5.1719 percent sit above the spot curve.
Matching periodicities for market reference rates
Before combining rates quoted on different bases, their compounding frequencies (periodicities) must agree. Suppose the three-month MRR is 1.25 percent and the six-month MRR is 1.75 percent, and we want the three-month rate three months forward, IFR3m,3m. The three-month rate compounds four times a year and the six-month rate twice, so the annual percentage rates must be aligned first.
Converting the 1.75 percent semiannual rate to a quarterly basis gives a quarterly APR of 1.74619 percent. Feeding that in, (1 plus 0.0125 over 4) times (1 plus IFR3m,3m over 4) equals (1 plus 0.0174619 over 4) squared, which solves to IFR3m,3m equal to 2.24299 percent. As a check, CNY100,000,000 invested at 1.25 percent for three months and rolled at 2.24299 percent, and the same sum invested at 1.75 percent for six months, both grow to CNY100,875,000.
A forward rate agreement (FRA) is an over-the-counter contract in which two parties agree to apply a set interest rate to a future period. Its underlying is a notional deposit imagined for a future window, to be placed then at a market reference rate that is locked in today at inception. The buyer, or long, is the fixed-rate payer: it agrees to pay interest at the agreed fixed rate and to receive interest at the MRR that runs from period A to period B, with the floating rate set on or just before the settlement date at time A.
An FRA is really a single-period interest rate swap. As with a swap, only the net of the fixed and floating interest changes hands, and the notional is never exchanged, serving only to size the interest. The fixed rate on the FRA is the implied forward rate for that period, the no-arbitrage rate that leaves the contract worth zero to both sides at inception.
The settlement turns on the difference between the fixed rate (the implied forward rate) and the realized market reference rate. From the fixed-rate payer’s side the amount due at the end of the interest period is that difference times the notional times the day-count fraction.
One twist: an FRA is cash settled at the start of the interest period rather than the end, so the end-of-period amount is discounted back at the realized reference rate to the settlement date.
From the earlier work, the implied three-month forward MRR beginning three months out (IFR3m,3m) is 2.24299 percent. Yangzi Bank enters an FRA as the fixed-rate payer on a CNY100,000,000 notional, paying 2.24299 percent on a quarterly actual/360 basis and receiving three-month CNY MRR. The bank uses the FRA to hedge a liability three months out on which it will owe MRR.
FRAs are used almost entirely by dealers and other financial intermediaries to hedge rate-sensitive assets and liabilities on the balance sheet. Together with the single-period swap that settles at the end of the interest period, the FRA is a basic building block for the interest rate swaps that issuers and investors rely on to manage rate risk.